When should one prefer using Total Variational Divergence over KL divergence in RL
I did not read those two specified linked/cited papers and I am not currently familiar with the total variation distance, but I think I can answer some of your questions, given that I am reasonably familiar with the KL divergence.

When you compute the $D_{KL}$ between two polices, what does that tell you about them

The KL divergence is a measure of &quot;distance&quot; (or divergence, as the name suggests) between two probability distributions (i.e. probability measures) or probability densities. In reinforcement learning, (stochastic) policies are probability distributions. For example, in the case your Markov decision process (MDP) has a discrete set of actions, then your policy can be denoted as $$\pi(a \mid s),$$which is the conditional probability distribution over all possible actions, given a specific state $s$. Hence, the KL divergence is a natural measure of how two policies are similar or different.
There are 4 properties of the KL divergence that you always need to keep in mind

It is asymmetric, i.e., in general, $D_{KL}(q, p) \neq D_{KL}(p, q)$ (where $p$ and $q$ are p.d.s); consequently, the KL divergence cannot be a metric (because metrics are symmetric!)
It is always non-negative
It is zero when $p = q$.
It is unbounded, i.e. it can be arbitrarily large; so, in other words, two probability distributions can be infinitely different, which may not be very intuitive: in fact, in the past, I used the KL divergence and, because of this property, it wasn't always clear how I should interpret the KL divergence (but this may also be due to my not extremely solid understanding of this measure).


and how is it different from what a $D_{TV}$ between the same two policies tells you?

$D_{TV}$ is also a measure of the distance between two probability distributions, but it is bounded, specifically, in the range $[0, 1]$ [1]. This property may be useful in some circumstances (which ones?). In any case, the fact that it lies in the range $[0, 1]$ potentially makes its interpretation more intuitive. More precisely, if you know the maximum and minimum values that a measure can give you, you can have a better idea of the relative difference between probability distributions. For instance, imagine that you have p.d.s $q$, $p$ and $p'$. If you compute $D_{TV}(q, p)$ and $D_{TV}(q, p')$, you can have a sense (in terms of percentage) of how much $p'$ and $p$ differ with respect to $q$.
The choice between $D_{TV}$ and $D_{KL}$ is probably motivated by their specific properties (and it will probably depend on a case by case basis, and I expect the authors of the research papers to motivate the usage of a specific measure/metric). However, keep in mind that there is not always a closed-form solution not even to calculate the KL divergence, so you may need to approximate it (e.g. by sampling: note that the KL divergence is defined as an expectation/integral so you can approximate it with a sampling technique). So, this (computability and/or approximability) may also be a parameter to take into account when choosing one over the other.
By the way, I think that your definition of the total variational divergence is wrong, although the DTV is related to the DKL, specifically, as follows [1]
\begin{align}
D_{TV} \leq \sqrt{\frac{1}{2} D_{KL}}
\end{align}
So the DTV is bounded by the KL divergence. Given that the KL divergence is unbounded (e.g. it can take very big values, such as 600k, this bound should be very loose).
Take a look at the paper On choosing and bounding probability metrics
 (2002, by Alison L. Gibbs and  Francis Edward Su) or this book for information about $D_{TV}$ (and other measures/metrics).